Hodge Structures on Posets

Let P be a poset with unique minimal and maximal elements 0̂ and 1̂. For each r, let Cr(P ) be the vector space spanned by r-chains from 0̂ to 1̂ in P . We define the notion of a Hodge structure on P which consists of a local action of Sr+1 on Cr, for each r, such that the boundary map ∂r : Cr → Cr−1 intertwines the actions of Sr+1 and Sr according to a certain condition. We show that if P has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of Hr(P ) into r Hodge pieces. We consider the case where P is Bn,k, the poset of subsets of {1, 2, . . . , n} with cardinality divisible by k (k is fixed, and n is a multiple of k). We prove a remarkable formula which relates the characters Bn,k of Sn acting on the Hodge pieces of the homologies of the Bn,k to the characters of Sn acting on the homologies of the posets of partitions with every block size divisible by k. 1. Hodge structures Let P be a finite poset with unique minimal and maximal elements 0̂ and 1̂. We use the notation (x1, . . . , xr) to denote r-chains 0̂ < x1 < x2 < · · · < xr < 1̂. In this case, we will sometimes write x0 and xr+1 for 0̂ and 1̂, respectively. In this paper we will only consider ranked posets, and we will write rk(x) for the rank of x ∈ P . For each r let Cr be a complex vector space with a basis consisting of r-chains in P . Define dj : Cr → Cr−1 by dj(x1, . . . , xr) = (x1, . . . , xj−1, xj+1, . . . , xr). Definition 1.1. A local action of Sr+1 on Cr is an action of Sr+1 on Cr such that for each j ∈ {1, 2, . . . , r}, (j, j + 1) · (x0, x1, . . . , xr, xr+1) = ∑ xj−1<z<xj+1 cz(x0, . . . , xj−1, z, xj+1, . . . , xr+1) for suitable constants cz. The idea of local actions on chain spaces of posets originated with Stanley (see [6]). Received by the editors December 12, 2001 and, in revised form, January 5, 2005. 2000 Mathematics Subject Classification. Primary 05E25. This work was supported in part by the National Science Foundation under Grant No. DMS0073785. c ©2006 American Mathematical Society Reverts to public domain 28 years from publication